Bounds for Rayleigh-Bénard convection between free-slip boundaries with an imposed heat flux

Abstract

We prove the first rigorous bound on the heat transfer for three-dimensional Rayleigh-Bénard convection of finite-Prandtl-number fluids between free-slip boundaries with an imposed heat flux. Using the auxiliary functional method with a quadratic functional, which is equivalent to the background method, we prove that the Nusselt number $Nu$ is bounded by $Nu\le 0.5999R^{1/3}$ uniformly in the Prandtl number, where $R$ is the Rayleigh number based on the imposed heat flux. In terms of the Rayleigh number based on the mean vertical temperature drop, $Ra$, we obtain $Nu\le 0.4646R{a}^{1/2}$. The scaling with Rayleigh number is the same as that of bounds obtained with no-slip isothermal, free-slip isothermal, and no-slip fixed flux boundaries, and numerical optimization of the bound suggests that it cannot be improved within our bounding framework. Contrary to the two-dimensional case, therefore, the $Ra$-dependence of rigorous upper bounds on the heat transfer obtained with the background method for three-dimensional Rayleigh-Bénard convection is insensitive to both the thermal and the velocity boundary conditions.

5 pages, 3 figures. Version 2: changed format to JFM, minor rewriting following comments by referees, added Appendix A

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